A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
2. Let’s see how much you
remember
What is a function?
How do you find rate of change of a function?
How do you graph a function?
How can you know by looking at the graph something is a
function?
How do you find domain and range of a function?
3. Functions
A function is a set of ordered pairs of numbers (x,y) in
which no two distinct ordered pairs have the same first
number.
Write the definition of a function in your own words
The domain of a function is the set of all possible x values
or the input.
The range is the resulting values y, or the output, from the
given input values.
How do you know whether x or y is independent or
dependent?
4. Rate of change
The rate of change of a function is found by dividing the outputs by the
inputs. Two points are needed to calculate the rate of change.
Later on another way to calculate the rate of change is to use the difference
quotient.
The difference quotient is
(f(x+h)-f(x))/h
Or
(fb-f(a))/b-a
How is the difference quotient similar to the slope formula?
5. Graphs of Functions
If f is a function, then the graph of f is the set of all points
(x,y) in the plane R^2 for which (x,y) is an ordered pair in f.
The Vertical Line Test determines whether a graph
represents a function.
Explain why the VLT test can show whether a graph is a
function.
7. Domain Restrictions
Domain restrictions are when the input values of a
function are restricted to certain values. Determine the
domain restrictions for the following functions and explain
your reasoning.
sqrt(x)
1/x
log(x)
8. General functions
Work with a partner to find the domain and range of the
following functions and graphs:
a) f(x)=x g) f(x)=1/x
b) f(x)=x^2 h) f(x)=log(x)
c) f(x)=x^3 i) f(x)=e^x
d) f(x)=sqrt(x) j) f(x)=sin(x)
e) f(x)={ 1-x if x<=1 k) f(x)=cos(x)
x^2 if x>1
9. Types of functions
A function f is an even function if for every x in the domain
of f, f(-x)=f(x)
A function f is an odd function if for every x in the domain
of f, f(-x)=-f(x)
Given two functions f and g, the composite function,
denoted by (f°g)(x)=f(g(x))
and domain of f°g is the set of all numbers x in the domain of
g such that g(x) is in the domain of f.
10. Exercises
Prove whether the following functions are even, odd, or
neither.
1) f(x)=10x^3-4x^2+3x+8
2) f(x)=-7x^7-x^3+5x
3) f(x)=x^3-x^2-1
4) f(x)=2x^2-3
11. Table of signs
The table of signs is created by looking at the signs of
parts of the function to see the overall change of the sign
of the function.
We look at the intervals of graph that are positive and
negative.
We can find the maximum and minimum values when we
look at the table of signs.
A minimum is when the sign of the graph changes
negative to a positive.
A maximum is when the signs changes positive to
negative.
12. For the function x^3+4x^2+x-6,
the table of signs is
Functio
n
-3 -2 1
x-1
- - - +
x+2
- - + +
x+3
- + + +
f(x)
- +
- +
Does the function have a max and/or a min? Give your reasoning.
13. Horizontal and Vertical asymptotes
To find the horizontal asymptote of rational functions in the
form
f(x)=(ax^n+…)/(bx^m+…)
If n<m, then y=0 is the horizontal asymptote
If n=m the the horizontal asymptote is y=a/b
If n>m, then there no horizontal asymptote but an oblique
asymptote.
The vertical asymptote is found by setting the denominator
equal to zero.
14. Find the horizontal and vertical
asymptote
1) f(x)=x^2+3x+1/4x^2-9
2) f(x)=x^2-x-2/x-2
3) f(x)=6x^2-3x+4
4) f(x)=x-12/2x^3+5x-3
15. Review
What is a function? Give an example of a function and a
non-function.
When does a function have a rate of change of zero?
When is the slope undefined?
What is the domain and the range of the function
f(x)=sqrt(x^2-1)
The function y=x+1 changes from negative to positive at
the point x=-1. What does that indicate?
